A period-doubling bifurcation corresponds to the creation or destruction of a periodic orbit with double the period of the original orbit. These bifurcations are especially prominent in the theory of one-dimensional, noninvertible maps, i.e.,dynamical systems that are actions of the semi-group $\Z^+$ on the unit interval, where infinite cascades of period-doubling bifurcations are typical, and exhibit certain universal properties.

Period doubling bifurcations, with a view on universal cascades

Local period doubling bifurcations in a nutshell

Formal presentation of our framework

We will only consider here dynamical systems that correspond to the action of the groups $\Z$ and $\R$, or the semi-groups $\Z^+$ and $\R^+$, on some phase space. For any such semi-group or group $G$ and any $x$ in the phase space, $O(x)=\{\phi_t(x)\}_{t\in G}$ is the orbit of $x$. Sometimes we refer to an orbit simply as $O$; it is then implicit that $O$ is $O(x)$ for any of the points $x\in O$, which happens for any orbit for a group action but only for periodic orbits (to be discussed next) for actions of semi-groups (orbits that are indeed special both for groups and for semi-groups actions).

We often deal with a sequence $O_0$, $O_1$, $O_2 ,\, \dots\, O_k ,\dots$ of such orbits of particular interest to us. We also consider parameter-dependent orbits $O_\mu$ (or $O_{k,\mu}$, $k=0,1,\ldots$ if we are dealing with sequences of orbits), where $\mu$ varies in some parameter space or portion thereof. Since the (semi) group $G$ carries a linear order compatible with its structure, we may talk about periodic orbits, i.e., orbits such that $O(x)=\{\phi_t (x)\}_{t\in G}=\{ \phi_t(x)\}_{t\in K}$ where $K$ is an interval of elements of $G$ that can be replaced by any translate of itself. The smallest length of such a $K$ is the minimal period, $\tau_{\min}$, of the periodic orbit (or the period (of said periodic orbit)), and any integer multiple $\tau$ of $\tau_{\min}$ is called a period (of said periodic orbit). The periods of a periodic orbit are characterized by the fact that $\phi_\tau(O)=O$, point-wise, i.e., $\phi_\tau(x)=x$ for all points $x$ of $O$. In what follows, by period we mean the period, except otherwise specified. This convention is useful when dealing with bifurcations, i.e., changes in the topological structure of the set of orbits.

When $G$ is $\Z$ or $\Z^+$ (respectively $\R$ or $\R^+$), a periodic orbit either has a finite number of points (respectively, is an embedded circle). In any case, it is a compact
object that is invariant under the action of $G$, that is: it is just forward-invariant if $G$ is one of the semi-groups $\Z^+$ and $\R^+$ and it is
fully invariant (i.e., forward and backward invariant) if $G$ is one of the groups $\Z$ and $\R$.

These four cases for $G$ are all of interest when studying the period doubling bifurcation, but a major role will be played by the case $G = \Z^+$; it corresponds to the iteration of noninvertible maps.

Dynamics defined by iteration of a continuous map

For the dynamics defined by iteration of a continuous family $f_\mu$, a period-doubling is a "bifurcation" whereby a $\tau$-periodic orbit $O_{0,\mu}$ looses its 'stability as the parameter $\mu$ crosses the critical value $\mu_c$ of $\mu$ (we will assume, w.l.o.g. that $\mu$ crosses from below), and at which point either

a stable $2\tau$-periodic orbit emerges (supercritical period doubling); or

an unstable $2\tau$-periodic orbit coalesces with $O_{0,\mu}$ and is destroyed (subcritical period doubling).

From this point on we assume that the families of dynamical systems are sufficiently smooth---see Remark 1 below---to rule out some pathologies that
make the picture more complicated. We will also assume that the families and
their members are generic except otherwise specified. For example, for generic semi-flows the period of an orbit
varies continuously with the parameter $\mu$.

Universal cascades of period doubling bifurcations

In order to emphasize the special status of period doubling among other local bifurcations, we will briefly review what has made the development of our understanding of cascades of period doubling (bifurcations) so important in the overall development of smooth dynamical systems theory and its many applications to mathematics, sciences, and technology.

The local theory of period doubling looks like any other local bifurcation theory, e.g., if one takes the second iterate in the case of a map, the period doubling bifurcation gets replaced by a non-generic pitchfork bifurcation. Period doubling is however quite special because it is often met in models for natural phenomena in infinite cascades of period doubling bifurcations that have metric universality properties that resemble what is observed in second order phase transitions.

Remark 1. Throughout this article, $C^3$ is enough smoothness for whatever is stated about one-dimensional dynamics. Often much less is required but the smoothness requirement for universality depends on the dimension (Gambaudo and Tresser 1992). An important
consequence of Universality as seen in period doubling cascades is that, while the difference between any two smoothness classes lies under the atomic scale (indeed under any finite scale), high enough smoothness has macroscopic, hence quite visible consequences.

Consider a one parameter family of smooth enough unimodal interval maps $f_\mu$, hence maps with a single critical point, say 0 assumed to be a maximum up to change of variable. By genericity $f^"_\mu$ is bounded away from 0 near $x=0$ so that, writing $f_\mu(x)=f_\mu(0)-a_\mu |x|^\beta+h.o.t., $
the exponent $\beta$ of $f_\mu$ is 2. We will also informally comment on maps with exponent $\beta\geq 1$. For every natural number $i\in \{0,1,2,\dots\}$, let:

$\mu_i$ be the (largest) parameter value such that 0 belongs to a periodic orbit of period $2^i$ of $f_{\mu_i}$ while $f_{\mu_i}$ has no periodic orbit of period $2^{i+1}$, (with more words one could also use the period doubling bifurcations between $\mu_i$ and $\mu_{i+1}$);

$\overline{\mu}_i$ be the (smallest) parameter value such that $f_{\overline{\mu}_i}$ has topological entropy $h(f_{\overline{\mu}_i})=\frac{\ln 2}{2^{i+1}}$ where the topological entropy $h(f_\mu)$ describes how much chaos there is in the dynamics of $f_\mu$.

Setting $\delta_i={\mu_i}-{\mu_{i-1}}$ and $\overline{\delta}_i={\overline{\mu}_{i-1}}-{\overline{\mu}_{i}}$ one has:
\[ \tag{1}
\mu_\infty=\lim_{i\to\infty} \mu_i=\lim_{i\to\infty} \overline{\mu}_i\,,
\]
an equality that puts the boundary of topological chaos defined as $h(f_\mu)>0$ at $\mu_\infty$. Some deep theorems now yield Eq.(1) under mild conditions: see (Melo and Strien, 1993) for an extensive bibliography and a broad coverage of one-dimensional dynamics.

Conjecture 1. [Parameter-space Universality 1]
One has (Feigenbaum, 1978; Coullet and Tresser, 1978; Tresser and Coullet, 1978)
\[ \tag{2} \frac{\Delta_{i}}{\Delta_{i+1}}\to\delta\equiv
\delta(2)=4.669201609101990\dots\,. \]
The critical exponent $\nu(\beta)=\frac{\ln 2}{\ln{\delta (\beta)}}$ allows us to write
$2^i\propto(\mu_\infty-\mu_i)^{-\nu(\beta)}$, where the period length $2^i$ appears as a correlation length when $\mu<\mu_c$, and there is a $\nu(\beta)$ for any exponent $\beta$ with $1< \beta$.

The equality $\nu(\beta)=\nu'(\beta)$ for $\beta>1$ is called a scaling law in statistical mechanics.

Conjecture 3. [Phase-space Universality]
The map $f_{\mu_\infty}$ has a measure theoretic attractor that is a Cantor set whose small scale geometry is asymptotically universal (Feigenbaum, 1978; Coullet and Tresser, 1978; Tresser and Coullet, 1978), with the ratios of the Cantor set accumulating in a universal Cantor set of ratios (see (Birkhoff et al., 2003) and references therein).

Like in the case of second order phase transitions, the explanation of the universality
phenomenology was proposed in terms of a Renormalization Group (or RG) theory, leading to another conjecture about how some RG procedure should explain the universal behavior observed on one-dimensional maps (Feigenbaum, 1978; Coullet and Tresser, 1978; Tresser and Coullet, 1978; and other dynamical systems as initially reported in (Coullet and Tresser, 1978) ).

Such considerations lead to the following renormalization conjecture. Consider the two renormalization operators, ${\mathcal{R}}_1$ acting near the critical point and ${\mathcal{R}}_{0}$ acting near the critical value, where ${\mathcal{R}}_{1}$ and ${\mathcal{R}}_{0}$ associate to a map $g=f_\mu$ with $\mu\in (\mu_0, \overline{\mu}_0]$ a
rescaled version of the restriction of $g^2$ respectively to $[x_1,x_0]$ and $[x_0, x_2]$ where $x_i=g^{-i}(x_0)$ and $x_1<x_0< x_2$. With enough smoothness, the ${\mathcal{R}}_{j}$'s can be applied $i+1$ times for $\mu\in (\mu_i, \overline{\mu}_i]$, and infinitely many times on $f_{\mu_\infty}$.

Conjecture 4. [Renormalization]
If the exponent of $f_\mu$ is 2, there exists a codimension 1 unstable fixed point for any of two Renormalization operators with the same unique unstable eigenvalue, $\delta$, that accounts for the behavior described in Conjectures 1 and 3 (Feigenbaum, 1978; Coullet and Tresser, 1978; Tresser and Coullet, 1978) and in Conjecture 2 (Tresser and Coullet, 1978). Furthermore, the stable manifold of these fixed points contains all smooth enough unimodal maps with a generic, quadratic, critical point. For other exponents, one has to restrict to the proper class of maps or the renormalization fixed points have more directions of instability.

Combining these conjectures with further numerical experiments and results from asymptotic methods and Poincaré maps applied to differential equations (ODE's and PDE's) also led to the following conjecture (Coullet and Tresser, 1978).

Conjecture 5. [Applicability and observability]
Cascades of period doubling with the generic $\delta$ also happen in the transition to chaos in families of dissipative diffeomorphisms of the plane, whence, using well known techniques, in families of differential equations and in the natural phenomena and engineering settings that they model closely enough (Coullet and Tresser, 1978).

In particular, Coullet and Tresser each told Libchaber in February 1978 that a universal period doubling cascade was to be expected in the convection experiments that Libchaber and Maurer were conducting at the Ecole Normale Supérieure, in a setting where one period doubling was already reported by Libchaber: Libchaber and Maurer
reported the observation of the universal period doubling cascade in 1980 (Libchaber and Maurer, 1980).

Progresses toward these conjectures for one dimensional maps have been considerable following a major breakthrough by Sullivan (1992) (see in particular Avila and Lyubich, 2011; De Carvalho et al. 2005, Davie, 1999; de Faria et al., 2006; Lyubich, 1999; McMullen, 1994; Sullivan, 1992) although many questions remain wide open. The global real analytic theory of Sullivan, Lyubich, McMullen, Avila, etc and the smooth generalization of Davie, de Faria-de Melo-Pinto followed an earlier computer assisted local theory by Lanford and the theory for the exponent $\beta=1+\epsilon\,$ by Collet, Eckmann, and Lanford that used perturbation methods that could not cover $\epsilon$ close to 1. The universality theory for dissipative embeddings of the disk and higher dimensional maps is still in its infancy (De Carvalh et al., 2005), driven by Lyubich, Martens and collaborators after early steps by Collet-Eckmann-koch and Gambaudo-van Stiren-Tresser who indicated the first connection of universality with the difficulties in proving a Closing Lemma in smoothness class $C^2$ and above.

Topological Universality and other precursors and asides

Before metric universality was discovered, combinatorial and topological forms of universality were described for real maps, for instance in a theorem by Sharkovskii (1964) that remained too long mostly unknown (see also Metropolis (1973); Milnor and Thurston (1988)), and RG-related ad hoc theories were formulated to understand some of that (Gumowski and Mira, 1975; Derrida et al., 1978).

The first detailed study of the cascade of period doubling
bifurcations was made on quadratic maps and already used a computer and references to dynamics in $\C$ (see Myrberg (1963) and references therein). Many people later contributed to various aspects of 1-dimensional dynamics, from combinatorics and topology to measure theory. It was recognized that Nielsen-Thurston's classification of homeomorphisms of surfaces contained the seeds of a discussion of topological universality in 2-dimensional
dynamics. For maps on 2-manifolds that are smooth enough, no chaos can appear in a one parameter family without cascades of period doubling (Yorke and Alligood, 1985) but, although no cascade is needed before the transition in that setting, Tresser has conjectured that there is a cascade before any transition to chaos for smooth enough families of uniformly area contracting embeddings of the 2-disk. First steps toward that have been made by Gambaudo,
Tresser, etc.. Due to a result by Bloch and Hart, $C^1$ is enough smoothness to prove that a full period doubling cascade must precede chaos on the interval. Such $C^1$ cascades have no metric universal properties.

Several other examples of universality, topological and/or metric, and associated RG's were later discovered in maps in one and more dimensions, an early example being (Derrida et al., 1978). Among these, only period doubling cascades yield a
codimension-1 transition to chaos, but the relevance of universality and RG goes way beyond the transition to chaos. Strongly asymmetric critical points first studied by Arnéodo-Coullet-Tresser and other hypotheses of the piecewise type will not be considered here; some may consider that we have included all that in the list of pathologies, but the main reason why we keep mute on that and many other sub-topics is the explosion of findings and results on matters related to period doubling and cascades of such events.

Period doubling bifurcations seen on signals

Consider a dynamical system. We call signal a segment of orbit of a measure theoretic attractor, expressed as vectorial function of time, or one of the coordinates in general position as a scalar function of time so that some partial embedding of the attractor can
be reconstituted from that one dimensional time dependent signal. Such signals are easily piecewise approximated (“piecewise" because of numerical noise) in numerical simulation and essentially correspond to (reasonably) asymptotic time signals segments that can be extracted from actual experiments. There might be several coexisting signals corresponding to different measure-theoretic attractors. We are particularly interested in the cases when some typical signal $X_{0,\mu}(t)$ that is periodic with period $\tau(X_{0,\mu}(t))$ when $\mu\in [\mu_c-\epsilon, \mu _c]$ turns to a signal $X_{1,\mu}(t)$ periodic with period $\tau(X_{1,\mu}(t))$ when $\mu\in (\mu _c, \mu_c+\epsilon]$ with
\[ \tag{4} \lim_{\mu\to \mu_c^+} (\tau(X_{1,\mu}(t)))\;=\;2\cdot\lim_ {\mu\to \mu_c^-} (\tau(X_{0,\mu}(t)))\ . \]
Here $\epsilon>0$ is assumed to be small enough to guaranty that no bifurcation occurs in $[\mu_c-\epsilon, \mu _c)$ nor in $(\mu _c, \mu_c+\epsilon].$

The period $\tau(X_{1,\mu}(t))$ is simply twice $\tau(X_{0,\mu}(t))$ (i.e., no need for the limits when $\mu$ approaches $\mu_c$ in Eq.(4)) in the case of a map. For flows and semi-flows the period $\tau(X_{1,\mu}(t))$ is $2$ with respect to the
Poincaré map to a local cross section of the flow near $X_{0,\mu}(t)$, and in all cases, equation ((4)) expresses what period doubling means in the case when an orbit is created at $\mu=\mu_c$.

Coalescence as $\mu\to\mu_c^-$ means that the two signals
collide as the parameter $\mu$ reaches $\mu_c$ from below, value when they vanish. In the period doubling case an unstable $O_0(\mu)$ remains above $\mu_c$ while, e.g., the
coalescence in the destructive saddle-node bifurcation leaves no periodic orbit.

We will mostly consider maps as one can reduce to that case by using some Poincaré map, and only consider fixed points till Sec. 1.12 as iteration of a map lets one reduce to that case as detailed in Sec. 1.12. Thus except otherwise specified, $\tau(X_{0,\mu}(t))=1$, meaning that $X_{0,\mu}(t)$ is a fixed point of some map $f_\mu$.

Dependence upon the dimension

A consequence of smoothness is that the loss of stability allowing the period doubling to happen can be read off the spectrum (set of eigenvalues) of the map (the Poincaré map when considering the dynamics of a flow or semi-flow) linearized around the fixed point that bifurcates. Indeed, the characterization of a bifurcation taking place (or of an instability, in the language of applied mathematics) is a parameter-dependent eigenvalue whose norm crosses 1 from below. For period doubling, the crossing of the unit circle is at -1 and happens along the real line. One can also say that for period doubling the eigenvalue crosses -1 “from above". To simplify the discussion one assumes that all crossings of the unit circle in the spectrum are made at non-zero speed, i.e., the value 1 is not a
stationary norm value for any eigenvalue as a function of the parameters.

In dimension 1, $\lambda_\mu=f'_\mu(x_0)$, the unique eigenvalue of the map linearized near $x_0$, crosses -1 from above so that $|\lambda_\mu|-1$ increases and $x_0$ becomes more and more repelling; $\lambda_\mu<-1$ tells us that nearby orbits go away from $x_0$ in an alternating way (\ie $(f_\mu(x)-x_0)\cdot(x-x_0)<0$ for $x\not=x_0$) in a small enough
neighborhood of $x_0$. Generically the dynamics is described by the normal form (6) after suitable change of variables.

In dimension greater than 1 an eigenvalue of the linearized map crosses -1 from above. Then, if all other eigenvalues of the maps linearized near the bifurcating fixed point have norm well separated from 1, for $\mu$ close enough to $\mu_c$ there is a $\mu$-dependent one-dimensional central manifold (or center manifold).

For such matters see for instance Carr (1981), Kelly (1967), Vanderbauwhed (1989) and references therein, both in the pure and applied mathematics literatures and Haragus and Iooss (2009) and references therein for infinite dimension. A central manifold is locally unique but not necessarily globally unique and is not necessarily as smooth as the flow or the diffeomorphism that leaves it invariant. Any of the central manifolds is tangent to the eigenvector for the eigenvalue that crosses -1 and generically supports the dynamics described by the normal form (6) after suitable change of variables. This holds true as long as all the other elements of the spectrum remain far enough from the unit circle. We always make this hypothesis here in order to avoid getting in the more complex world of the bifurcations that occur when many parameters are varied at once (bifurcations of codimension two or higher).

For bifurcation problems rather than speaking of “a parameter dependent central manifold" one may prefer to consider the parameters as extra (dynamically trivial) coordinates along which the eigenvalues of the linearized flow are trivially equal to 1. Since only one eigenvalue gets out of the unit circle (through -1) in the case of the period doubling bifurcation, one parameter often suffice so that we end up with a two dimensional central manifold with one dimension here to let one capture the evolution of the local map in a neighborhood of $(x,\mu)=(x_0,\mu_c)$.

Period doubling: a one-dimensional phenomenon for maps

Figure 1: Period doubling for maps: evolution with parameters with continuous lines representing stable orbits and bigger gaps between dots meaning less dimensions of stability.

Assuming enough smoothness, one can use normal forms (expressions that are simplified as much as possible by changes of variables) that leave on a central manifold of dimension 1 not counting the dimension of the parameter (see Sec. 1.5 ).

For a map in one dimension, in the neighborhood of $(O_0, \mu_c)=(0,1)$, the generic normal form for the period doubling bifurcation of the fixed point set at $0$ reads:
\[ \tag{6} f_\mu(x)= -\mu\cdot x \pm x^3+ \mathrm{h.o.t.} \]
By the very fact that a bifurcation happens at $\mu_c$, $O_0(\mu_c)$ is linearly marginal (or (linearly) marginally stable), and we have the following Stability-based Criterion.

If $O_0(\mu_c)$ is nonlinearly (or NL) unstable, the bifurcation is subcritical.

If $O_0(\mu_c)$ is NL stable, the bifurcation is supercritical.

From this criterion or from a direct computation one deduces from (6) that:

($+$) If the $+$ sign is chosen, the bifurcation is supercritical with the evolution of the dynamics as the parameter crosses $\mu_c$ from below described in the discussion of equation (4).

($-$) If the $-$ sign is chosen, the bifurcation is subcritical with the evolution of the dynamics as the parameter crosses $\mu_c$ from below described in the discussion of equation (5).

Remark 2. The validity of (6) as a normal form requires $f_\mu$ to be invertible on the interval. In particular, (6) cannot contain the next period doubling in a meaningful way. Besides, any symmetry (say $x\mapsto-x\,$ for (6)) trivially forces a symmetry breaking bifurcation to happen before the next period doubling.

Figure 2: Period doubling for differential equations: description for two parameters, one below, one above the bifurcation with continuous lines representing stable orbits and bigger gaps between dots meaning fewer dimensions of stability.

Local analysis for maps in dimension 1

In dimension 1, the central manifold is simply the local part of the full phase space near the bifurcating orbit. It remains to compute the criticality type of the bifurcation i.e., whether the bifurcation is supercritical or subcritical; the degenerate case that
separates supercriticality from subcriticality will be considered in Sec. 1.8.

To get the most general formula, we assume that $f_{\mu_c}(x_0)=x_0$ (the fixed point property) and that $f_{\mu_c}'(x_0)=-1$ (the period doubling bifurcation's first order condition). In what follows, as usual, $\epsilon$ has a positive but arbitrary small absolute value. Note that comparing the distances $|f_{\mu_c}(x_0+\epsilon)-x_0|$ and
$|\epsilon|$ is not of any help when investigating the non-linear stability property of the
point $x_0$ under $f_{\mu_c}$. Indeed the signs of $\epsilon$ and $f_{\mu_c}(x_0+\epsilon)-x_0$ are opposite for $\epsilon$ small enough because the slope $f'_{\mu_c}$ is $-1$. Consequently we need to compare $\epsilon$ to $f^2(x_0+\epsilon)-x_0$. Assuming $f_\mu$ to be three times differentiable for $(\mu, x)$ near $(\mu_c,x_0)$, a straightforward computation then yields:
\[ \tag{7} f^2_{\mu_c}(x_0+\epsilon)-x_0=\epsilon+\epsilon^3 \left(-\frac{1}{2}(f''^2_{\mu_c}(x_0))^2- \frac{1}{3}f'''_{\mu_c}(x_0)\right)\, \]
that leads one to define:
\[ \tag{8} \mathcal{Q}f(x)\;=\; -\frac{1}{2}(f''(x))^2-\frac{1}{3}f'''(x)\, \]
Then, using the phenomenological Stability-based Criterion, one deduces from ((8)) a computable criterion that goes back at least to (Allwright, 1978):

If $\mathcal{Q}f_{\mu_c}(x_0)<0$, then $x_0$ is NL stable and the bifurcation is supercritical.

If $\mathcal{Q}f_{\mu_c}(x_0)\,>\,0$, then $x_0$ is NL unstable and the bifurcation is subcritical.

The quantity $\mathcal{Q}f_{\mu_c}(x_0)$ may be zero, and higher order terms may be necessary to determine the stability property of $x_0$. It may even happen that the bifurcation be marginal at all orders, in which case, if no flat term breaks the degeneracy, the fixed point $x_0\,=\,f_{\mu_c}(x_0)$ is the common end point of two
segments of period two points that are exchanged by $f_{\mu_c}$.
Then $ f_{\mu_c}(x_0+\epsilon)= x_0-\epsilon $
so that
\[ \tag{9} f_{\mu_c}^2 (x_0+\epsilon)= x_0 +\epsilon\, \]
for $x=x_0+\epsilon$ close enough to $x_0$.

Figure 3: Period doubling from supercritical to subcritical through the degenerate marginal case that can provide all configuration by deformation of the period doubling orbits curve: only the most three basic cases have been represented here.

We draw in Figure 3 a classical bifurcation diagram with the parameter $\mu$ (increasing when the period doubling bifurcation takes place) along the abscissa and $x$ along the ordinate. We use a heavy line for stable objects, dotted lines to represent unstable objects, and smaller dots and smaller gaps for marginal objects. We see that there is always a line emanating on both sides from the point $(\mu_c, x_0)$, that represents the orbits that goes from stable to unstable. The line representing a marginal object is vertical when one is in the degenerate, affine case. Moreover, the marginal line:

Bends forward with a parabolic shape as a heavy line close enough to $(\mu_c, x_0)$ when one removes the degeneracy so that the bifurcation is supercritical;

Bends backwards with a parabolic shape as a doted line close enough to $(\mu_c, x_0)$ when one removes the degeneracy so that the bifurcation is subcritical.

The continuation of the (doted) lines is no longer a local problem; for global consideration on single period doubling, see Sander and Yorke (2011) and earlier references therein (similar work was done many years before for the Poincaré-Andronov-Hopf bifurcation by Alexander, Mallet-Paret, Nussbaum, Yorke etc.). The growth of the size of the period-doubled orbit (as a function of $|\mu-\mu _c|$) fails to be proportional to $\sqrt{|\mu-\mu _c|}$ ({\it which represents the generic but not the general case}) if and only if there is no third order term.

Local theories are only local

In all issues regarding the bifurcation diagram, including issues of size of bifurcated cycles, it is crucial to assume that $|\mu-\mu _c|$ is “small enough", where the required smallness depends in a highly non-universal way on the structure of the evolution equations (map or vector field). There are cases where the very initial growth of the size of the bifurcated cycle (the distance to the cycle that has just bifurcated) starts indeed like $\sqrt{|\mu-\mu _c|}$ (or another growth in non-generic cases), but then beyond some $\mu'\not=\mu_c$, a tiny increase of the parameter provokes an explosive increase of the size of the period doubled cycle. This phenomenon was studied in the context of the
Poincaré-Andronov-Hopf bifurcation for flows in dimension 2, where it took the French name of “canard".

The frequent miracle of asymptotics

Despite the possibility of canards-like effects, the normal form often has a very large domain of validity, extending often by far the provable domain of validity, a phenomenon that enables the success of such methods in the description of many natural phenomena.

From loss of stability to more generally one less stable direction

We have first considered bifurcations associated to loss of stability. This is only the most basic and perhaps the most important case. More generally the orbit $O_0(\mu)$ does not need to be stable at the beginning of the process if the phase space has dimension at
least 2. In general said orbit $O_0(\mu)$ looses one direction of stability in the process and:

In the subcritical case the bifurcation destroys an orbit $O_1(\mu)$ that has one direction of stability less than what the original orbit $O_0(\mu)$ has before the bifurcation.

In the supercritical case the bifurcation produces an orbit $O_1(\mu)$ that has as many directions of stability as the original orbit $O_0(\mu)$ has before the bifurcation.

Period doubling bifurcations from any period

So far we have mostly dealt with the period doubling bifurcation of a fixed point of $f_\mu$, but this is as much generality as one ever needs for studying local period doubling bifurcations since a periodic point of period $\tau$ of $f_\mu$ is a fixed point of $f^\tau_\mu$ (the periodic orbit of length $\tau$ is made of $\tau$ fixed points of
$f_\mu^\tau$). By the chain rules, $f^\tau_\mu$ has the same slope at all these points, and more generally all $Df^\tau_\mu$'s have the same spectrum.

The Schwarzian derivative in the context of period doubling

The quantity $\mathcal{Q}f_{\mu_c}(x_0)$, whose sign determines the criticality type of the doubling bifurcation (Allwrigh, 1978), is proportional to the Schwarzian derivative of $f_{\mu_c}$ evaluated at $x_0$ (Singer, 1978). We discuss the negative Schwarzian derivative
condition in this Section; not only does this condition play a crucial role in deep issues such as understanding precursors to Universality and the question of how smoothness that depends on arbitrary small scale data can have visible effects, but in the context of period doubling, it guaranties that $\mathcal{Q}f^p_{\mu}(x_0)$ is negative for all the period doubling bifurcations for any period $p$ as $\mu$ varies, hence all period doubling bifurcations are supercritical.

On a theorem by Élie Cartan

In dynamics, a cocycle is an operator that associates to a given map a function or tensor that satisfies a chain rule under composition. There are three useful cocycles in the study of the dynamics and the geometry of orbits of a one-dimensional map $g$. They
are:

A theorem by Cartan (1937) states that these are the only possible cocycles in dimension 1. Their respective kernels are the translation group, the affine group, and the projective group in dimension 1. Here we only consider the Schwarzian derivative.

For any three-times differentiable map $g$ the Schwarzian derivative of $g$ reads:
\[ \tag{10}
Sg(x)=\frac{g'''(x)}{g'(x)}-\frac{3}{2}\left(\frac{g''(x)}{g'(x)}\right)^2\,.
\]
We then notice that since $f_{\mu_c}'(x_c)=-1$, we have $ Sf_{\mu_c}(x_c)=9\mathcal{Q}f_{\mu_c}(x_c) $ so that the criterion $\mathcal{Q}f_{\mu_c}(x_c)<0$
that we had for superstability of the period doubling bifurcation in dimension 1 can be expressed as:
\[\tag{11}
Sf_{\mu_c}(x_c)<0\,.
\]
The important point here is that if $Sf_\mu<0$ on the interval where $f_\mu$ acts as an endomorphism (a continuous map from that interval to itself) then for any value of $\mu$ in the parameter range $S(f_\mu^n)<0$ for all $n$, i.e., all the iterates of $f_\mu$ have negative Schwarzian derivative. In particular, if the $f_\mu$'s are $C^3$ maps with $Sf<0$ for all values of $\mu$ in some parameter range, then all periodic points that get unstable by period doubling do so the supercritical way as $\mu$ varies.

Singer's Theorem and the first developments

These basic properties of maps $f$ with $Sf<0$ were pointed out in 1978 by Singer (1978), who also proved the following result about unimodal $C^3$ maps $f$ with $Sf<0$, also known as S-maps.

Theorem 1 (Singer, (1978)). If $f: I\to I$ is a S-map, then the immediate basin of attraction of any attracting periodic point contains either a critical point of $f$ or a boundary point of $I$. Furthermore, any neutral periodic point $x_0$ of $f$ (i.e., $|(f^n)'(x_0)|=1$ for $x_0$ such that $f^n(x_0)=x_0$) that is not on the boundary of $I$ is attracting, and there are no intervals of periodic points.

After Singer published this result (coincidentally (?) in the same journal as Allwright (1978)) many researchers, including Collet, Eckmann, Guckenheimer, and Misiurewicz, used it in studies of several questions, mostly about families of interval maps. These were soon followed by the next wave of smooth interval maps researchers, in fact with serious overlap: de Melo, Nowicki, van Strien, Sullivan, Blokh, Lyubich, and their students. Many properties of S-maps were later generalized to multimodal maps: see in particular van Strien and Vargas (2004,2007).

As was immediately recognized, the problem with the $Sf<0$ condition is that it is not general and that it is not even invariant under all smooth changes of coordinate. It is thus important to notice that, with enough smoothness, subcriticality is exceptional in the sense that only finitely many subcritical bifurcations can happen for any iterate for maps that are $C^3$ (and in fact a bit less). This was proved by Kozlovski (2000), following work, e.g., by Martens, de Melo, van Strien, and Sullivan (1988), and pioneering work by Guckenheimer (1987) for application to renormalization. In the context of renormalization, sometimes it is possible to prove that the $Sf<0$ property emerges after enough renormalizations (see for instance de Faria and de Melo (1999, 2000) and de Faria et al. (2006) ).

The macroscopic meaning of $Sf<0$

The theorem of Singer stated above can be regarded as an analogue of a classical result in holomorphic dynamics due to P. Fatou, which states that the basin of attraction of an attracting periodic point of a degree $d\geq 2$ rational map always contains a critical point (leading to the conclusion that there are only fintey many such attracting cycles). This suggests that $S$-maps behave somewhat as complex one-dimensional maps. A further hint at this behavior is provided by the following considerations.

The $Sf<0$ property is special for a number of reasons, one of which is the following fact.

Lemma 1.If $g$ is three times differentiable, then the property $Sg<0$ is equivalent to saying that the function $\sqrt{\frac{1}{|g'|}}$ is convex in each interval where $g'$ does not vanish.

This implies the following result:

Proposition 1.If $Sg$ is negative on an interval $I$, then $|g'|$ has no non-zero
local minimum in $I$.

This last property of the derivative of a map with negative Schwarzian derivative is linked to a macroscopic property of S-maps that we will state here. For that purpose, we need to
recall the definitions of two cross ratios.

Given an interval $T$ (for “total") and a subinterval $M$ (for “middle part"), so that $T\setminus M$ is made up of a “left part" $L$ and a “right part" $R$, and denoting by $|I|$ the length of an interval $I$, we consider two cross ratios, namely
\[\tag{12}
a(M,T)=\frac{|M|\cdot|T|}{|L\,\cup\,M|\cdot|M\,\cup\,R|},
\]
and
\[ \tag{13}
b(M,T)=\frac{|M|\cdot|T|}{|L|\cdot|R|}.
\]
Then, if $f$ is a map that is a diffeomorphism from $T$ to its image in $\R$, let us write
\[ \tag{14}
A(f, M,T)=\frac{a(f(M),f(T))}{a(M,T)},
\]
and also
\[\tag{15}
B(f, M,T)=\frac{b(f(M),f(T))}{b(M,T)}.
\]
These ratios measure the distortions of the corresponding cross-ratios under $f$.

Proposition 2.We have $A(f, M,T)\geq1$ and $B(f, M,T)\geq 1$ whenever $f$ is an
S-map that is a diffeomorphism from $T$ onto its image in $\R$.

Figure 4: Negative and positive Schwarzian maps and their effects on cross-ratios.

This expansion of cross-ratios -- which is extremely useful in establishing real a-priori bounds in many contexts, including period-doubling renormalization -- is akin to the fact that a rational map of the Riemann sphere always expands the hyperbolic (Poincaré) metric in the complement of its post-critical set. In order to enhance this analogy,
we make the following further considerations.

First note that the Schwarzian derivative of a $C^3$ diffeomorphism $f:T\to T^*$ between two (open) intervals $T,T^*$ can be recast in the following form: \[ Sf(x)\;=\; \lim_{y\to x} \frac{\partial^2}{\partial x\partial y}\log{\left(\frac{f(x)-f(y)}{x-y}\right)}. \]

Next, following Sullivan, define the Poincaré density of an open interval $T=(a,b)$ to be \[\rho_T(x)\;=\;\frac{(b-a)}{(x-a)(b-x)}\] This is the infinitesimal form of the Poincaré metric on $I$. Thus, the Poincaré length of $M=(c,d)\subseteq I$ is \[ \ell_T(M)\;=\;\int_M\, \rho_T(x)\,dx\;=\;\log{(1+b(M,T))} \] where $b(M,T)=(b-a)(d-c)/(c-a)(b-d)$ is the cross-ratio of the four points $a, b, c, d$, as introduced earlier.

The derivative of $f:T\to T^*$ measured with respect to the Poincaré metrics in $T$ and $T^*$, namely \[ D_Tf(x)\;=\;f'(x)\,\frac{\rho_{T^*}(f(x))}{\rho_T(x)} \] is called the Poincaré distortion of $f$. It is identically equal to one if $f$ is Möbius, in which case $f$ preserves cross-ratios. Now consider the symmetric function $\delta_f: T\times T\to \mathbb{R}$ given by \[
\delta_f(x,y)\;=\;\left\{
\begin{array}{ll}
\displaystyle{\log\,\frac{f(x)-f(y)}{x-y}}
\, , &\ x\neq y, \\
{}&{}\\
\displaystyle{\log\,{f'(x)}} \, , & \ x=y \ .
\end{array}\right.\] Then an easy calculation shows that \[ \tag{16} \log\,D_Tf(x)\;=\;\delta_f(x,x)-\delta_f(a,x)-\delta_f(x,b)+\delta_f(a,b).\]

When $f$ is $C^3$ its Poincaré distortion is controlled by the second order mixed derivative of $\delta_f$, since in that case \[ \log\,D_Tf(t)\;=\; - \int\!\int_Q\,\frac{\partial^2}{\partial x\,\partial y}\delta_f(x,y)\,dx\,dy \ , \] where $Q$ is the rectangle $(a,t)\times (t,b)$. Moreover, when $(x,y)\to (t,t)$ the integrand above becomes $6\,Sf(x)$, where $Sf$ is the Schwarzian derivative of $f$. This is consistent with the fact that maps with negative Schwarzian increase the Poincaré metric and consequently expand cross-ratios (Figure 4 4).

The negative Schwarzian, from hypothesis to conclusion

The following theorem of Kozlovski illustrates the difference between high smoothness and low smoothness dynamics, something that somehow culminates with Universality phenomena which appear in period doubling and other types of recurrence for smooth enough maps.
What is important here is that cross ratios can also be defined for macroscopic intervals, a first clue for the visibility of high enough smoothness and its universal consequences.

Theorem 2. (Kozlovski (2000)) Let $f : X \to X$ be a $C^3$ unimodal map of an interval to itself with a non-flat non-periodic critical point $c$. Then there exists an interval $J$ around the critical value $f(c)$ such that if $f^n(x) \in J$ for $x\in X$ and $n > 0$ then $Sf^n(x) < 0$.

The negative Schwarzian property is therefore quite ubiquitous in the small (for $C^3$ maps), at least near non-flat critical points. A precursor to the above theorem is the fact that every sufficiently deep renormalization (around the critical point) of an infinitely renormalizable, $C^3$ one-dimensional map with a power-law critical point has negative
Schwarzian (see de Faria and de Melo (1999), de Faria et al., (2006) ). This fact has important consequences for period-doubling (and more general combinatorics) in the form of asymptotically universal bounds on the geometry of the post-critical set. For example, for infinitely renormalizable unimodal maps which are symmetric about the critical point, it implies that, of all the intervals that one sees at a sufficiently deep level of renormalization, the one containing the critical point is the largest (this was first proved by Guckenheimer (1979); see also de Fari (2006) [p. 760] ). For the consequences of not imposing the symmetry hypothesis at the accumulation of a period doubling cascade, see Chandramuoli et al., (2009).

For a general reference on period doubling, cascades and renormalization, covering also the $Sf<0$ property, see de Melo and van Strien (1993) and keep posted for the next edition of that book.